Optimal Screening by Risk-Averse Principals�

Suren Basovy Xiangkang Yinz

May 2009

Abstract

This paper studies the e¤ects of principal�s risk aversion on principal-agent

relationship under hidden information. It �nds that the agent�s equilibrium

e¤ort increases and approaches the e¢ cient level as the principal�s risk aversion

increases and tends to in�nity. Allowing for random participation by the agent,

his e¤ort is likely to be e¢ cient even the principal�s risk aversion is �nite. For

the case of common agency with random participation, it is optimal for the

principals to make the agent the residual claimant on pro�ts and the principals�

net pro�ts monotonically decrease to zero when their risk aversion tends in�nite.

Keywords: principal-agent model, risk aversion, random participation, com-

mon agency

JEL Classi�cation Numbers: D42, D43, D82.

�The authors are grateful to Georgy Artemov, the participants at Economic Theory Workshop inMelbourne University and seminar in La Trobe University for helpful suggestions. They also thankBuly Cardak for editorial assistance.

ySchool of Economics and Finance, La Trobe University, Bundoora 3086, Victoria, Australia.Email: s.basov@latrobe.edu.au

zSchool of Economics and Finance, La Trobe University, Bundoora 3086, Victoria, Australia.Email: x.yin@latrobe.edu.au

1

1 Introduction

This paper studies systematically the e¤ects of the risk aversion of principals on

the optimal screening contracts. Existing literature in screening by nonlinear payment

schedules, originating in papers of Rothschild and Stiglitz (1976), Mussa and Rosen

(1978), and Maskin and Riley (1984), typically considers a certainty environment

or assumes that principals are risk neutral and maximize expected payo¤s. Though

several papers and textbook treatments use a formulation of the screening problem

that allows for risk aversion on the principal�s side,1 most of them do not study

systematically the e¤ects of principal�s risk aversion on the equilibrium allocation.

A notable exception is Gence-Creux (2000), who studies a joint problem of adverse

selection and moral hazard in a regulation contract with a risk-averse principal and

two di¤erent types of agents.

The reason for this omission in the theory development may be due to the recogni-

tion that the population of agents is su¢ ciently large and the e¤ects of uncertainty are

negligible, although the risk aversion of principals is well accepted as a more plausi-

ble and more realistic assumption than risk neutrality. Uncertainty and risk attitude,

however, become crucial when a principal faces a small �nite number of agents with

hidden information. For example, at the time when a �rm hires a CEO the �rm may

not know exactly or is not certain about the CEO�s ability of pro�t-making and work

ethics. Consequently, the �rm�s pro�ts under the CEO�s management are uncertain.

Such uncertainty to principals is also quite obvious when a principal, such as a small

1See, for example, Fudenberg and Tirole (1992) for a textbook treatment, and papers by Anglinand Arnott (1991), Spier (1992), Page (1997), Gence-Creux (2000), and Celik (2003).

2

municipality, signs a regulatory contract with a private "public service" operator,

such as a large multinational �rm (Gence-Creux (2000), La¤ont and Rochet (1998)),

because the transfer from the principal to the operator depends on the unknown char-

acteristics of the agent such as operating costs. In �nancial market, when a liquidity

provider (principal) posts a limit order on an electronic trading system, she does not

know the size of next market order (agent) she will meet although she can impose

a limit on the maximum amount of buying or selling (Biais, Martimort and Rochet

(2000)). Under these circumstances, the payo¤ to a principal from the use of a par-

ticular contracting mechanism is uncertain and therefore the optimal mechanism will

be a¤ected by the degree of risk aversion of the principal. This paper focuses on the

role played by the risk aversion of principal(s) and its e¤ect on the optimal screening

and on the equilibrium e¤ort exerted by the agent.

The hidden information under the consideration, which is summarized by the

type of an agent, may include his valuation for services he can exchange with the

principal(s) as well as his valuation of outside opportunities. For concreteness, we

investigate the case, where the principal hires the agent to perform a task for her. To

focus on screening, we abstract from the moral hazard issue by assuming that work

e¤ort is contractible by observing the output of the agent through a deterministic

function of e¤ort, but the cost of e¤ort is private information to the agent. In Section

2, we consider a scenario where the set of possible types (i.e., marginal costs of

e¤ort) forms an interval on the Euclidean line but the value of the outside option

is constant and known to the principal. Thus, in the spirit of Mussa and Rosen

(1978), we develop a model of bilateral contracting, which can accommodate the

3

risk aversion on the principal side. It is found that risk aversion makes pooling at

equilibrium less likely. More importantly, the equilibrium e¤ort provision increases

in the degree of the principal�s risk aversion under certain relatively non-restrictive

conditions. When the principal becomes su¢ ciently risk averse, the e¤ort reaches the

e¢ cient level. Total payment and marginal payment to the agent increase as well

when the principal becomes more risk averse. What are the driving forces behind

these results? If the principal is risk neutral, her goal is to maximize the expected

pro�ts, net of compensation to the agent. However, if she is risk averse, the volatility

of net pro�ts, including the risk of breaking down the principal-agent relationship,

matters too. Thus, she is willing to pay the agent more in exchange for a lower

volatility, which also stimulates the agent to work harder as he faces a higher and

steeper pay. In the extreme, the principal�s degree of risk aversion is in�nitely high

and she wants a non-random net pro�t. Then, she can optimally designs a contract

to let the agent to be residual claimant. In turn, the agent will exert an e¢ cient

level of work e¤ort. This section ends with a closed-form optimal wage designed by

a risk-averse principal in a special case of the model.

Section 3 extends the model in Section 2 by allowing the agent�s private value

of the outside option to be random but independent of his marginal cost of e¤ort,

paralleling the Rochet and Stole�s (2002) extension of the Mussa-Rosen model (1978).

Therefore, the type is actually two-dimensional. However, since the value of outside

option does not a¤ect the marginal cost of e¤ort, screening is possible only on one

dimension: the cost of e¤ort. If there is only one principal, we �nd once again that

the equilibrium e¤ort exerted by the agent approaches the e¢ cient level when the

4

principal is su¢ ciently risk averse. Moreover, there is a distribution of participation

costs for which making the agent residual claimant on the pro�ts is the optimal

solution to the principal�s problem and the principal�s pro�ts decline in her coe¢ cient

of absolute risk aversion. In the case of common agency, making the agent residual

claimant on the pro�ts is optimal in a full-participation symmetric equilibrium, for

any distribution of participation costs. Similar to the case of single principal, the

net pro�ts of the principals monotonically decrease and converge to zero as their

coe¢ cient of absolute risk aversion tends to in�nity. However, there is an important

di¤erence between the single principal and common agency cases. In the common

agency case, the pro�ts left to the principals a¤ect only the distribution of the surplus

because the agent de�nitely works for a principal in equilibrium. But the agent may

fail to be hired in the single principal equilibrium so that a decrease in the principal�s

net pro�ts can improve the probability of the agent to be hired and in turn the

e¢ ciency of the economy.

Our analysis shows that the risk aversion on the principal(s)�side generally plays

a positive role in terms of improving e¢ ciency and the agents�welfare. This �nding

is similar to Rochet and Stole�s (2002) results on the positive e¤ects of random par-

ticipation on nonlinear pricing, although the economic mechanisms behind them are

di¤erent.

5

2 Agent with a known outside option

Consider a principal-agent relationship, where a risk-averse principal hires an

agent to perform a task for her. The task involves exerting e¤ort, e; which pro-

duces pro�ts �(e) for the principal,2 where �(�) is strictly increasing, strictly concave

function. Although she may not be able to directly observe the agent�s e¤ort she can

infer it uniquely from the pro�t generated by the agent. The agent�s preferences are

summarized by:

u(w; e) = w � et; (1)

where type, t; can be interpret as the marginal cost of the agent�s e¤ort. Although

t is the private information of the agent, the principal knows that t is drawn from

a distribution with a strictly positive probability density function, f(t); on (t1; t2);

where 0 � t1 < t2 � 1. By the Taxation Principle (Rochet (1985)) one can without

loss of generality restrict mechanisms used by the principal to be nonlinear wage

schedules, w(e).

For a given wage schedule w(�) : R! R; de�ne the agent�s surplus by:

s(t) � maxe(w(e)� et): (2)

Denote e(t) the utility-maximizing e¤ort of a type-t agent. By the envelope theorem,

we have

s0(t) = �e(t): (3)

2We abstract from moral hazard problem by assuming that �(�) is deterministic.

6

The principal�s expected utility is given by:

t2Zt1

V (�(e(t))� w(e(t)))f(t)dt;

where utility function V (�) is strictly increasing, concave, and twice di¤erentiable.

Thus, the principal�s problem is to choose a wage schedule, w(�), to maximize her

expected utility, subject to (3) and the implementability constraint, which states

that e(�) is non-increasing.

Let us �rst concentrate on the relaxed problem, i.e. drop the implementability

constraint. Then the monopolist solves

max

t2Zt1

V (�(e(t))� w(e(t)))f(t)d(t) (4)

s.t. s0(t) = �e(t); s(t2) = 0: (5)

Note, an agent is characterized by his marginal cost of e¤ort so that the principal can

always properly select the wage schedule making the surplus of the most ine¢ cient

agent, agent t2; equal to his outside option. In (5), the outside option has been

normalized to zero. The Hamiltonian for this problem is:

H � V (�(e)� s(t)� e(t)t)f(t)� �(t)e(t): (6)

The Pontryagin maximum principle states that if fe(�); s(�)g is the solution to the

principal�s problem then there exists �(t) 2 C1() such that

7

8>>>>>>>>>><>>>>>>>>>>:

s0(t) = �e(t)

�0(t) = V 0(�(e)� s(t)� e(t)t)f(t)

� = V 0(�(e)� s(t)� e(t)t)f(t)(�0(e)� t)

s(t2) = 0; �(t1) = 0

: (7)

In particular, the above system implies that

�0(e(t1)) = t1; (8)

i.e. the marginal pro�t of the principal is equal to the marginal cost of e¤ort exerted by

agent t1. In other words, we obtain the well-known result of no e¢ ciency distortion at

the top.3 On the other hand, the second equation of system (7) implies that �0(t) > 0

for all t 2 (t1; t2): Since �(t1) = 0; it is obvious that �(t) > 0 for all t 2 (t1; t2].

Recalling the third equation of system (7), the strict positivity of �(t) means

�0(e(t)) > t (9)

for all t 2 (t1; t2]. Thus, all agents, except for agent t1, underprovide e¤ort, no matter

in pooling or separating equilibrium.

Substituting the third equation of (7) into the second and after some rearrange-

ment one obtains:

d

dt((�0 � t)f � F ) = R(�0 � t)2fe0; (10)

3Note, t1 type (i.e. the type for whom the e¤ort is least costly) generates the highest totalsurplus. Therefore, we call it the top type.

8

where R � �V 00=V 0 is the Arrow-Pratt�s coe¢ cient of absolute risk aversion and F (�)

is the cumulative density function corresponding to f . Note, we have dropped the

variables in the functions to simplify notations and e0 � de(t)=dt: The initial condition

for (10) is (8). In the analysis below, we assume that R is constant, i.e. V (�) is a

CARA utility function,

V (x) =1� exp(�Rx)

R; (11)

with V (0) = 0 and limR!+0 V (x;R) = x: If the principal is risk neutral, then R = 0

and (10) implies

�0(e(t)) = v(t) � t+ F (t)f(t)

: (12)

Expression (12) is similar to the well-known characterization of the e¢ ciency distor-

tion by a risk-neutral principal (see Varian, 1989). It leads to the solution for the

complete problem provided that the virtual type, v(�); is non-decreasing. If the vir-

tual type is decreasing in some range of types the optimal solution will entail some

pooling. Note that if �(�) satis�es Inada condition at zero all agents will participate

in the contract. However, exclusion region will be non-empty if �0(0) < inft v(t): Our

�rst proposition establishes that pooling, if it happens, will occur for R 2 [0; R�) for

some R� � 0. In particular, if equilibrium is fully separating under risk neutrality

it will remain fully separating for any degree of risk aversion. More precisely, the

following proposition holds:

Proposition 1 Assume that the pooling region is non-empty for R = R0. Then, it

is also is non-empty for any R < R0.

9

Proof. With routine calculus, (10) can be written as

e0(t) =2f(t)� (�0(e)� t)f 0(t)[�00(e)�R(�0(e)� t)2]f(t) : (13)

Let er(t; R) denote the solution of the relaxed problem (4)-(5),4 it thus satis�es (13).

Then (8) and (9) imply that

�0(er(t; R)) � t; (14)

where equality holds only when t = t1: By the assumption of non-empty pooling

region for R = R0, there exists t = t� such that

e0(t�; R0) > 0.

Therefore, (13) yields

2f(t�)� (�0(er(t�; R0))� t)f 0(t�) < 0;

which requires f 0(t�) > 0 because of (14) and f(t�) > 0: It is straightforward to show

that Hamiltonian (6) is supermodular in (e;R) and in turn er(�; R) is increasing in

R. Since er(�; R) is increasing in R and �0(�) is decreasing, we have, for any R < R0,

that

2f(t�)� (�0(er(t�; R))� t)f 0(t�) < 0;

4Equilibrium e¤ort e(t) is implicitly determined by R. So we explicitly express the e¤ect of Rby e(t; R) when it is relevant. The same treatment is also applied to equilibrium wage, etc. Forsimplify, we slightly abuse notations and e0(t; R) below should be considered as @e(t; R)=@t:

10

which implies from (13) that

e0(t�; R) > 0,

i.e., there is pooling equilibrium at t� for all R < R0. Q.E.D.

Proposition 1 indicates that risk aversion on the principal side reduces the chance

of pooling equilibrium. Therefore, the condition ensuring separating equilibrium un-

der risk neutral also guarantees separating equilibrium under risk aversion as stated

by the following corollary.

Corollary. 1 If

f 0(t) � 2f 2(t)

F (t); (15)

then for all R � 0 the equilibrium is separating.

Proof. Condition (15) is equivalent to the requirement that v0(t) � 0; i.e. that

the virtual type is increasing. It guarantees that there is no pooling for R = 0 and

therefore, by Proposition 1, for any R > 0: Q.E.D.

It is worth to notice that condition (15) in the corollary does not impose a sig-

ni�cant restriction because the log-concavity of F (t) or the monotonicity of hazard

rate f(t)F (t), a widely adopted assumption in the literature (e.g. Biais, Martimort and

Rochet (2000)), requires f 0(t) � f2(t)F (t)

.

Proposition 2 Assume that f(t) satis�es (15) or the equilibrium is fully separating

11

for all R: Then, e(t; R) increases in R.

Proof. By Corollary 1, pooling has been ruled out, solution to (10) or equivalently

to (13) provides the solution to the complete problem. Now observe that e0(t) increases

in R for given t and e; and the value of e at t = t1 does not depend on R. The

Gronwall�s lemma (see, for example, Bellman, 1943) implies that e(t; R) is increasing

in R. Q.E.D.

Proposition 3 Under the assumptions of Proposition 2, the equilibrium supply of

e¤ort is e¢ cient when R tends to in�nity, i.e.

limR!+1

e(t; R) = eeff (t);

where the e¢ cient e¤ort, eeff (t); is determined by �0(eeff ) = t:

Proof. Rewrite equation (13) as

e0(t)(�00(e)

R� (�0(e)� t)2)f(t) = 2f(t)� (�0(e)� t)f 0(t)

R:

De�ne function G(t; e) = 2f(t)� (�0(e)� t)f 0(t): Since (t; e) 2 [t1; t2]� [0; e(t1)] = C;

where C � R2 is a compact set, there exists K > 0 such that 0 < G(t; e) < K

for all (t; e) 2 C. Taking into account that under the assumptions of Proposition 2,

e0(t) < 0; one obtains

limR!+1

�0(e(t; R)) = t;

which completes the proof. Q.E.D.

12

The intuition behind Propositions 2 and 3 are straightforward. If the principal

is risk neutral, what she concerns is the expected pro�ts, net of wage expenditure.

However, if she is risk averse, the volatility of net pro�t, �(e) � w(e); concerns her

too. Thus, she is willing to sacri�ce net pro�ts and pay the agent more to reduce the

volatility. The increased wage payment will stimulate the agent to work harder and

in turn the e¢ ciency is improved. The more risk averse is the principal, the higher

wage is she willing to pay, and in turn the greater e¤ort is exerted by the agent.

When R ! +1; the principal tries to avoid all pro�t uncertainty. This requires

choosing a wage schedule, which ensures that the net pro�t yielded from meeting any

agent is constant; that is, �(e) � w(e) = �0. Clearly, this strategy makes the agent

the residual claimant on the �rm�s pro�ts. Taking its derivative with respect to e

and recalling the �rst order condition for agent utility maximization w0(e) = t; we

immediately obtain �0(e(t)) = t; i.e., the e¢ cient e¤ort provision. Now we formalize

our claim on the e¤ects of risk aversion on the wage schedule.

Proposition 4 Under the assumptions of Proposition 2, both total wage w(e) and

marginal wage w0(e) increase in R; i.e. wR(e;R) > 0 and weR(e;R) � 0:

Proof. Given the agent�s surplus, the total wage can be written as w(e) =

inft(s(t) + et) (see, for example, Basov 2005). Now

wR(e;R) = sR =

t2Zt

eR(� ; R)d� > 0; (16)

13

where the �rst equality follows from the envelope theorem, the second equality is de-

rived from (3) using initial condition (5), and the last inequality follows from Propo-

sition 2.

For marginal wage, the �rst- and second-order conditions for the type-t agent who

faces wage schedule w(e;R) imply

8>><>>:we(e;R) = t

wee(e; R) � 0: (17)

Totally di¤erentiating the �rst equation in (17) with respect toR and using the second

inequality and Proposition 2 one obtains weR = �wee(e;R)eR � 0: Q.E.D.

It is clear that as the principal becomes more risk averse the agent is paid more for

the same e¤ort level, and therefore his welfare increases (see (16)). In other words,

risk aversion on the principal side actually makes the agent better o¤. Next, we

provide an example for the special case of the model. The explicit solution to the

example will illustrate the main results of our previous analysis.

Example. Let us assume that the production function is �(e) = e� e2

2and the type

is distributed uniformly on (0; 1): Then system (10) becomes:

e0(t) = � 2

1 +R(1� e� t)2 (18)

with initial condition e(0) = 1: Apparently, there is e0(t) < 0; i.e. any solution of this

di¤erential equation is decreasing in t. Therefore, there is no pooling at positive e¤ort

14

levels and (18) completely determines the solution of the problem. Let us introduce

q(t) by q(t) � 1� e(t); then

q0(t) =2

1 +R(t� q)2 (19)

and q(0) = 0: De�ne implicit function q�(t) by:

q�(t) � 1pRln1�

pR(t� q�(t))

1 +pR(t� q�(t))

: (20)

It is easy to check that (20) satis�es system (19). The corresponding e¤ort solves:

e(t) = 1� 1pRln1�

pR(t+ e(t)� 1)

1 +pR(t+ e(t)� 1)

:

Agents who choose the outside option are those whose type t falling in (0; t�(R));

where

t�(R) =(exp(

pR)� 1)p

R(1 + exp(pR))

:

Note that t�(R) is decreasing in R with t�(0) = 0:5 and limR!1 t�(R) = 0; i.e. as the

principal becomes more risk averse the chance that the agent takes the principal�s o¤er

approaches one, which agrees with our previous observation that the e¤ort provision

approaches the e¢ cient level.

To solve for the optimal wage, let us introduce variable m �pR(t+ e� 1): Then

equation (20) implies:

m = tanh

pR(e� 1)2

; (21)

15

where the hyperbolic tangent tanh(�) is de�ned by tanh y = exp(y)�exp(�y)exp(y)+exp(�y) : Substituting

the �rst order condition for the agent�s optimization w0(e) = t into (21), one obtains:

w0(e) = 1� e+tanh

pR(e�1)2p

R:

Finally, carrying out integration and taking into account the participation constraint

w(0) = 0; one obtains:

w(e) = �(e) +2

Rlncosh

pR(e�1)2

coshpR2

;

where the hyperbolic cosine, cosh(�); is de�ned by cosh y = exp(y)+exp(�y)2

:

3 Agents with random outside option

Now we are going to enrich the model of the previous section by assuming that

the value of the outside opportunity is unknown to the principal, i.e. it is a part of

the private information of the agent. Formally, the type of an agent is characterized

by (t; x) 2 (t1; t2)� (x1; x2) � R2. His utility, when he provides e¤ort e and receives

wage w; is:

U(t; e; w; x) = w � et� kx: (22)

We further assume that random variable x is independent of t and has a cumulative

density function N(�). To simplify the notation we will assume that k = 1 in the

single principal case of Subsection 3.1 but will restore it in the common agency case

16

of Subsection 3.2.

3.1 A single risk-averse principal

We �rst consider the case of a single risk-averse principal. The agent�s surplus

is still de�ned by (2). Then the probability that an agent accepts the contract and

provides a positive e¤ort is N(s): Therefore, the principal solves

max

t2Zt1

V (�(e)� s(t)� e(t)t)N(s(t))f(t)dt (23)

subject to (3). Note, the boundary condition s(t2) = 0 in (5) no longer holds because

the value of the outside option is uncertain to the principal. Therefore, it is an optimal

control problem with both ends being free. The Hamiltonian for this problem is:

H = V (� � s� et)N(s)f(t)� �(t)e(t) (24)

and the Pontryagin maximum principle implies:

8>>>>>><>>>>>>:�0(t) = V 0(� � s� et)N(s)f(t)� V (� � s� et)N 0(s)f(t)

V 0(� � s� et)(�0(e)� t)N(s)f(t)� �(t) = 0

�(t1) = �(t2) = 0

(25)

Proceeding in the same way as in the previous section, one obtains:

1

N

d

dt(N(�0 � t)f)� f = R(�0 � t)2e0f + N

0V

NV 0f: (26)

17

If x takes only one value x0 with certainty, then N(s) = 1 for s � x0 and equation

(26) is reduced to (10). The boundary conditions in (25) implies

N(s(ti))(�0(e(ti))� ti) = 0 for i = 1; 2: (27)

It is reasonable to only consider the case where agents participate in equilibrium

with a positive probability. Then, N(s(t1)) > 0 and �0(e(t1)) = t1; which is the

conventional �no distortion at the top�property. On the other hand, the boundary

condition at t2 implies either a type-t2 agent participates with zero probability or the

provision of his e¤ort is also e¢ cient. This result of either no-service or no-distortion

at the bottom does not depend on the risk aversion of the principal and has been

observed by Rochet and Stole (2002) too.

To investigate the e¤ects of risk aversion, let us assume a Bernoulli utility function

for the agent again. It is obvious that the �rst term on the right hand side of (26)

becomes dominant when R is su¢ ciently large. Therefore, it leads to an e¢ cient e¤ort

when R tends to in�nity. The formal proof of this conclusion is similar to Proposition

3 and is omitted. So we only state it as a corollary below.

Corollary 2 The results in Proposition 3 hold with random participation.

The possibility of both type-t1 and type-t2 agents exerting an e¢ cient e¤ort leads

us to ask under what condition all agents will exert an e¢ cient e¤ort when they

face a wage schedule proposed by a principal with �nite risk aversion. The e¢ ciency

requires

w(e) = �(e)� AM ; (28)

18

where AM is a constant, which can be interpreted as the net pro�ts to the principal.

SupposeN(x) is an exponential function thatN(x) = B exp(�x) with � > 0. Because

N(x) is a cdf on some interval (x1; x2); there are N(x1) = 0 and N(x2) = 1; which

require x1 = �1 and B = exp(��x2); respectively. In words, these conditions say

that the value of the outside option must be distributed on (�1; x2) according to cdf

N(x) = exp(�(x�x2)) for the equilibrium to be e¢ cient, conditional on participation.

Recalling w0(e) = t; substituting (28) into (26) yields:

V 0

V=

R

exp(RAM)� 1= �. (29)

Thus, we arrive at the following Proposition.

Proposition 5 A risk-averse principal imposes a two-part wage schedule (28) with

AM =1

Rln(1 +

R

�): (30)

if and only if the outside option is distributed according to the following cumulative

distribution function

N(x) = minf1; exp(�(x� x2))g: (31)

Proof. We have already proved that if the optimal wage has two parts as (28)

the distribution of the outside option should satisfy (31). The reverse follows from

log-concavity of distribution (31) (see, Rochet and Stole, (2002)). Finally, (30) follows

from (29). Q.E.D.

Note that while wage schedule (28) implies that the agent provides e¢ cient e¤ort

19

subject to participation, the participation decision is not e¢ cient as long as AM

is positive. However, AM in (30) decreases monotonically from 1=� to zero as R

increases from zero to in�nity, indicating that increase in the principal�s risk aversion

leads to e¢ ciency gains and, as R tends to in�nity, the optimal e¤ort converges to

the e¢ cient one.

It might be interesting to compare the results in Corollary 2 and Proposition 5

to the �ndings of Rochet and Stole (2002). They discovered that the introduction of

random participation can improve the allocative e¢ ciency in the sense that random

participation lifts the quality supplied to the consumer of each type and moves it

closer to the e¢ cient level. The �ndings in Corollary 2 and Proposition 5 take us

one step further and indicate that e¤ort provision actually is likely to be e¢ cient,

conditional on participation, when random participation combines with risk aversion

on the principal side.

3.2 Common agency

This subsection extends the analysis in the previous subsection by allowing the

agent to be hired by either of two principals, i.e. we have a common agency problem.

Both principals are assumed to have the same production function, �; and the same

constant coe¢ cient of absolute risk aversion, R; as we have speci�ed in the single

principal case. Let us consider a model in the spirit of Hotelling model, i.e. the two

principals are located at the ends of a linear city with a length equal to one, while the

agent is randomly located along the Hotelling line. Thus, we can interpret xi in (22)

as the distance of the agent from principal i, where superscript i 2 fL;Rg indexes

20

the principals, and k as the marginal transportation cost. The principal�s pro�t and

the agent�s e¤ort cost satisfy the following condition,

maxe(�(e)� te)) > k

2+ �; (32)

where � � 2kN(12)=N 0(1

2). Assumption (32) simply says that the total surplus, ignor-

ing transportation cost, is su¢ ciently big for positive gains of trade to exist, irrespec-

tive to the location of the agent on the Hotelling line. This guarantees the existence

of equilibrium at which the agent provides positive e¤ort with probability one.

Following (2), the agent�s surplus function from the relationship with principal i is

si(t) = maxe(wi(e)� te): Given the surplus functions, the probability that the type-t

agent takes the contract from the left end principal is:

QL(sL; sR) = N(minfsL(t)

k;1

2+sL(t)� sR(t)

2kg):

Therefore, given the right principal�s wage schedule; wR(e); and the surplus function,

sR(t); the left principal determines her wage strategy by maximizing her expected

utility:

max

t2Zt1

V (�(e)� sL(t)� te))QL(sL; sR)f(t)dt:

Clearly, the only di¤erence between this program and (23) is N(s) in (23) has been

replaced by QL(sL; sR). So, with given sR the �rst-order condition for the optimal

solution to the program is the same as (25) with proper notation changes. We would

like to concentrate on symmetric equilibrium.

21

Proposition 6 Let condition (32) hold. Then the necessary and su¢ cient conditions

for the existence of symmetric equilibrium in which the agent works for one of the

principals and supplies an e¢ cient e¤ort are that each principal adopts the following

wage schedule:

w(e) = �(e)� AD; (33)

where �xed fee AD is given by:

AD =1

Rln(1 + �R): (34)

Proof. Necessity: Without loss of generality we assume that the agent works for

the left principal. Since the provision of e¤ort is e¢ cient, there is �(t) = 0 for all

t 2 [t1; t2] from the corresponding second equation of (25). Thus, �0(t) = 0 and the

�rst equation in (25) implies

V@QL(sL; sR)

@sL= V 0QL(sL; sR): (35)

On the other hand, the �rst-order condition of agent�s optimization requires

w0(e) = t and e¢ ciency implies �0(e) = t: Therefore, the wage schedule must be

(33). To determine AD; we notice that

QL(sL; sR) = N(1

2+sL(t)� sR(t)

2k): (36)

22

Then, substituting (33) into (35) yields

1

�=

R

exp(RAD)� 1;

and in turn (34).

Su¢ ciency: There is �0(e) = t if both principals impose a wage schedule (33),

which means that the �rst-order conditions for both principals are satis�ed and the

equilibrium is e¢ cient. It remains to check that given (33) the agent will always

accept the contract from one of the principals. But, note that (34) implies that

AD decreases monotonically from � to zero as R increases from zero to in�nity, i.e.

AD � �. Hence, (32) ensures that the agent will will de�nitely participate, irrespective

to the principals�risk attitude. Q.E.D.

Proposition 6 states that e¢ cient wage schedule prevails with a random but full

participating common agent. This echoes the �nding that duopolists impose a cost-

based two-part tari¤ in nonlinear price competition by Armstrong and Vickers (2001),

and Rochet and Stole (2002). Although the risk aversion of the principals does not

a¤ect the e¢ ciency of the economy, it does change the income distribution. As (34)

shows, the pro�t of the principals, AD; declines in R and reach zero when R is in�nite.

Therefore, as the principals become more risk averse, the agent becomes better o¤.

23

4 Conclusions

This paper has developed a model that focuses on the e¤ects of principals�risk

aversion on the optimal work e¤ort supplied by an agent with an unobservable cost

of e¤ort. Although it features principals facing a single agent, extension to any �nite

set of agents is straightforward. While the model builds on the scenario of principals

hiring an agent, the theory and its results are directly applicable to other screening

problems of hidden information. We started by revisiting the standard Mussa-Rosen

(1978) model and showed that an increase in risk aversion on the principal�s side

monotonically pushes the provision of e¤ort toward the e¢ cient level. Intuitively,

this happens because risk aversion makes the principal put a greater weight on avoid-

ing pro�t uncertainty, particularly the worst possible scenario � a failure to make a

hiring. Risk aversion also reduces the chance that the contracting results in a pooling

equilibrium. We then extended the base model to allow for random participation,

paralleling Rochet and Stole�s (2002) extension to the Mussa-Rosen (1978) model.

We have established conditions for the optimal wage schedule leading the agent to be

the residual claimant on the pro�ts under both single principle and common agency

regimes. While under the common agency the conditions are quite general and simply

require the unit transportation cost is su¢ cient low to allow for full participation, the

e¢ cient wage schedule in the case of single principle require a quite speci�c distribu-

tion of participation costs. In both cases the net pro�ts of the principal(s) decrease in

their coe¢ cient of absolute risk aversion. Such a decrease improves e¢ ciency through

more e¢ cient participation decision in the case of a single principal, however, it only

a¤ects the distribution of rents between the principals and the agent in the case of

24

common agency. Generally, our analysis indicates that risk aversion on the principal�s

side plays a positive role in improving either productive e¢ ciency or agent�s welfare.

25

References

M. Armstrong and J. Vickers, 2001, Competitive price discrimination, Rand Jour-

nal of Economics, 32, 579-605.

P. M. Anglin, and R. Arnott, 1991, Residential real estate brokerage as a principal-

agent problem, Journal of Real Estate Finance and Economics, 4, 99-125.

S. Basov, 2005, Multidimensional Screening, Studies in Economic Theory, v. 22,

Berlin: Springer-Verlag.

B. Biais, D. Martimort and J.-C. Rochet, 2000, Competing mechanisms in a Com-

mon value environment, Econometrica, 68, 799-837.

R. Bellman, 1943, The stability of solutions of linear di¤erential equations, Duke

Mathematical Journal, 10, 643�647.

G. Celik, 2003, Mechanism design under collusion and risk aversion, UBC discus-

sion paper, No 03-14.

D. Fudenberg, and J. Tirole, 1992, Game theory, Cambridge, MA: MIT Press.

C. Gence-Creux, 2000, Regulation with a risk-averse principal, Contributed Pa-

pers, Econometric Society World Congress.

J.-J. La¤ont and J.-C. Rochet, 1998, Regulation of a risk averse �rm, Games and

Economics Behavior, 25, 149-173.

E. Masking and J. Riley, 1984, Monopoly with incomplete information, Rand

Journal of Economics, 15, 171-196.

M. Mussa, and S. Rosen, 1978, Monopoly and product quality, Journal of Eco-

nomic Theory, 18, 301-317.

26

F. H. Page, 1997, Optimal deterministic contracting mechanisms for principal

agent problems with moral hazard and adverse selection, Review of Economic Design,

3, 1-13.

J. C. Rochet, 1985, The taxation principle and multitime Hamilton-Jacobi equa-

tions, Journal of Mathematical Economics 14, 113-128.

J. C. Rochet, and L. Stole, 2002, Nonlinear pricing with random participation,

Review of Economic Studies, 69, 277-311.

M. Rothschild, and J. E. Stiglitz, 1976, Equilibrium in Competitive Insurance

Markets: An Essay in the Economics of Imperfect Information, Quarterly Journal of

Economics, 80, 629-649.

K. E. Spier, Incomplete contracts and signalling, 1992, Rand Journal of Eco-

nomics, 23, 432-443

H. R. Varian, 1989, Price discrimination, in Handbook of Industrial Organization

I, ed. by R. Schmalensee and R. D. Willig., New York and Oxford: North-Holland.

27